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Chapter 14
Moments and torque
14.1
M
Summary of moments and torques
Description
n
Expression
FQ /O
= r Q/O × FQ
i=1 MS/O
Moment of force FQ about point O Moment of a set S of forces about point O Shift theorem for moment of a force Torque is the moment of a set S of forces whose resultant is 0.
MS/O = MS/P = TS =
MF
Qi /O
+ r O/P × FS
MS/O
14.2
Moment of a vector
Q
FQ
The moment of a vector results from the cross product of a position vector with a bound vector.a The moment of the vector FQ (bound to point Q) about point O is Q denoted MF /O and is defined in terms of r Q/O (Q's position vector from O) as
a
r Q/O
O
(1)
A bound vector is a vector that is bound to a point.
MF
Q /O
= r Q/O × FQ
14.2.1
Moment arm of a vector about a point
Q
The moment arm of the bound vector FQ about point O is the distance d between O and the line of actiona of FQ and measures the effectiveness of FQ at creating a moment about O. This distance can be calculated in various ways, e.g, using the unit vector u in the direction of FQ and/or the angle between u and r Q/O . d =
a
FQ
d
r Q/O
O
r
Q/O
sin() =
r
Q/O
×u
Q
=
r Q/O · r Q/O

(r Q/O · u)2
Q
MF
Q /O
= FQ d
The line of action of the bound vector F
is the line passing through Q and parallel to F .
14.2.2
Moment of a set of vectors
MS/O = MS/O =
n i=1
The moment of a set S of bound vectors FQ1 , ..., FQn about a point O is defined as the sum of the moments of each bound vector in equation (2). In view of Newton's law of action/reaction in Section 13.5.5, the moment of a system S's internal forces is 0 and MS/O can be written much more simply as solely the moment of external forces on S. 141
MF
Qi /O
(2) (3)
(13.7)
MS/O
external
14.2.3
Statics, dynamics, and the moment of a set of forces
The moment MS/O of all forces on a system S about an arbitrary point O is related to the timederivative of S's angular momentum about O in N and other quantities. This relationships simplifies to static equilibrium in equation (4) when the system S is at rest (not moving) in N or S is considered massless.
O
MS/O
(statics)
=
0
(4)
14.2.4
Shift theorem for the moment of a set of vectors
MS/P , the moment of a set S of bound vectors about a point P , can be calculated in terms of: · MS/O , the moment of S about a point O · r O/P , O's position vector from P · FS , the resultant of S
P
r O/P
O
(5)
MS/P = MS/O + r O/P × FS
14.2.5
Torque of a set of vectors
Torque is the moment of a set S of vectors whose resultant is zero. TS = MS/O
where
FS = 0
and point O is any point
(6)
Since a couple is a set of vectors whose resultant (sum) is 0, a torque is the moment of a couple.1 A couple has a special property, namely, the moment of a couple about a point O is equal to the moment of the couple about any other point Q. As a result, a torque is not associated with a point. The following example highlights the difference between a torque and a moment. Consider the various sets S of forces in Figure ??, and fill in the following table by calculating FS , the resultant of S, and MS/O , MS/P , MS/Q , the moments of S about points O, P , and Q, respectively. Express your results in terms of the righthanded orthogonal unit vectors nx , ny , nz . S A B C D FS 10 ny MS/O 50 nz MS/P 0 MS/Q MS/O = MS/P = MS/Q ? Yes/No Yes/No Yes/No Yes/No Moment is a torque? Yes/No Yes/No Yes/No Yes/No
14.2.6
Torque on a rigid body (or reference frame)
Torques can be associated with reference frames when the vectors that comprise the torque have a special character. The notation TA is used to designate a torque of a couple whose vectors F1 , ..., Fn have lines of actions that pass through points Q1 , ..., Qn , respectively, each of which is fixed in a reference frame A. When Q1 , ..., Qn are also fixed in a second reference frame, e.g., B, then one designates the torque TA/B .
Torques are a special type of Moment, namely the moment of a couple. All Torques are Moments, but not all Moments are Torques. A useful analogy is all Toyotas are Motorvehicles, but not all Motorvehicles are Toyotas.
Copyright c 19922009 by Paul Mitiguy
1
142
Chapter 14:
Moments and torque
10 n
10 n
A
O
5m
B
P
3m
Q
O
5m
P
3m
Q
ny
10 n
10 n
C
O
6n 5m
nz
Q
6n
nx
10 n
D
O
5m 4n
P
3m
P
3m 6n
Q
Figure 14.1: Various sets of forces
14.3
Proof of shift theorem for the moment of a set of bound vectors
To establish the validity of equation (5), consider a set S of bound vectors F1 , ..., Fn , whose lines of action pass through points Qi (i = 1, ..., n), respectively. The moment of S about a point P is defined in terms of r Qi /P (Qi 's position vector from P ) as MS/P =
n i=1
(2)
r Qi /P × FQi
(7)
Qi 's position vector from P may be written as r Qi /P = r O/P + r Qi /O Substituting equation (8) into equation (7), distributing the summation, and rearrangement yields MS/P
n (7,8)
(8)
= =
r O/P × FQi +
n i=1
n i=1 n i=1
r Qi /O × FQi r Qi /O × FQi (9)
i=1
r O/P ×
FQ i +
The first summation in equation (9) is the definition of FS (the resultant of S). The second summation in equation (9) is the definition of MS/O (the moment of S about O). Combining these two facts produces equation (5), the shift theorem for the moment of a set of bound vectors, namely MS/P = r O/P × FS + MS/O
Copyright c 19922009 by Paul Mitiguy
143
Chapter 14:
Moments and torque
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